Machine control



Aug. 28, 1962 Filed Nov. 7, 1955 G. G. FAYARD MACHINE CONTROL FIGJL.

TYPU

13 Sheets-Sheet 1 INVENTOR Georges G. Fuyo rd Aug. 28, 1962 G. G. FAYARD 3,051,389

MACHINE CONTROL Filed Nov. 7, 1955 13 Sheets-Sheet 2 INVENTOR 7 Georges G. Fuyurd BY M M M BW (1 4 ATTQ R NEYS G. G. FAYARD MACHINE CONTROL Aug. 28, 1962 Filed Nov. 7, 1955 13 Sheets-Sheet 3 lNVENTOR Georges G. Fuyord ATTORNEYS Aug. 28, 1962 G. G. FAYARD MACHINE CONTROL 13 Sheets-Sheet 4 Filed Nov. 7, 1955 72A OH V 7:

INVENTOR.

GEORGES G.FAYARD BY /%m ATTORNEY Aug. 28, 1962 G. G. FAYARD 3,051,389

MACHINE CONTROL Filed Nov. 7, 1955 13 Sheets-Sheet 5 FIG.7

mvsmon Georges G. Fuyurd ATTORNEYS M if .n s L m m U m For m M Aug. 28, 1962 G. G. FAYARD MACHINE CONTROL Filed Nov. 7, 1955 INVENTOR Georges G. Fayord ATTORNEYS Aug. 28, 1962 Filed Nov. '7, 1955 G. G. FAYARD 3,051,389

MACHINE CONTROL 13 Sheets-Sheet 8 F I G -11 5in .f\... 1 1

flak) F e r F l 6. l2

INVENTOR L ,l/ n-1 G eerges G. Foyurd 1 M M M ATTORNEYS 1962 G. G. FAYARD 3,051,389

MACHINE CONTROL Filed Nov. 7, 1955 13 Sheets-Sheet 9 9D 12D 15D 18D 210 I Q 30 l 24m 270 500 550 as lfindc rees 30 6D 2IU INVENTOR Georges G. Fcyurd BY MWWQu-,$M T% ATTORNEYS Aug. 28, 1962 e. G. FAYARD ,3

MACHINE CONTROL Filed Nov. 7, 1955 13 Sheets-Sheet l0 FIG.14

INVENTOR Georges G. Fuyord BY 1% MM B 7275,

ATTORNEYS 1962 G. G. FAYARD 3,051,389

MACHINE CONTROL Filed Nov. 7, 1955 13 Sheets-Sheet 11 7 FIG -18 3D 5D 9D 120 150 180 210 240 270 SUD 55B 360 d INVENTOR Georges G. Foyurd f w'q M M /law v-f yb AT TORN EYS Aug. 28, 1962 G. G. FAYARD 3,051,389

MACHINE CONTROL Filed Nov. 7, 1955 13 Sheets-Sheet 12 INVENI'OR GEORGES G FAYARD I ill I. lll ld l uwm lfl mum-.7524

mom B F5050 23% r n 55:. $2559 Unite .tes

The present invention relates to automatically controlled machines, for example machines for the shaping of turbine blades and the like.

The invention provides a machine of this type which dispenses with memory devices for the shapes of patterns such as cams to be followed by a feeler or magnetic or perforated tapes which record a function of space for reproduction by the machine. Instead the machine requires only that the curve to be followed by the cutting tool be known in the form of a parametric equation, or alternatively that the coordinates of a finite number of points on the curve be known.

In general terms a turbine blade or a compressor or propeller blade can be considered as being made up of a plurality of right cylinders having altitudes of any desired degree of smallness stacked together. The shape and the dimensions of the right sections of these cylinders, hereinafter to be called profiles, generally varies from one cylinder or cylindrical element of the blade to the next. Of course it is also possible for the shape and dimensions of the successive profiles to remain constant, being simply displaced angularly with respect to each other.

The path followed by the cutter in milling a profile is a closed curve parallel to the profile, and it will hereinafter be referred to as the parallel curve. The profile itself may for example be of convex cylindrical type of two sections.

This parallel curve can usually be defined in Cartesian or polar parametric coordinates which can be developed in the form of a Fourier series limited to a certain number of terms. A complete description of the curve is obtained by causing the parameter to vary between zero and 211-. If the curve is defined by p points, the number of terms 2n of the limited series is equal to the number p of points which are given for representation of the parallel curve in question.

Prior attempts have been made to control machine tools by control units generating signals representative of the abscissa and the ordinate of a plane profile to be machined or of its parallel curve through Fourier analysis and synthesis. The parameter chosen has been, so far as I am aware, the polar angle of a point on the profile. This has the advantage of making it possible to drive by the same movement the pattern and workpiece support and the analyzer and synthesizer of the control unit. But practice has shown that, for hydrodynamic and aerodynamic profiles, the convergence of the Fourier series in terms of such a parameter is rather poor.

I have found that, by a proper choice of the angular parameter in terms of which the Cartesian or polar coordinates of the parallel curve are expressed, it is possible for a given limit of accuracy to obtain developments having good convergence and a small number of terms. The invention is based on this fact. For a given curve there are in general an infinite number of possible developments in Cartesian or polar coordinates.

The invention is equally applicable to the production of workpieces whose profiles are not closed such as the shaping of grooves or special cams. It is sufiicient in such cases to join the ends of the desired profile curve atnt ice

by a suitable contour and to operate the command mechanism for the machine tool provided by the invention only between particular values of the parameter pertaining to the profile desired to be achieved while disabling it for other values of the parametre. Similarly the invention is applicable to the machining in part or whole of dies intended for stamping of metal sheets such as those used in automobile bodies. In such applications the invention obviates the use of the elaborate plaster models hitherto employed with a feeler for machining of dies in the automobile body industry.

The desired developments of the profile coordinates into limited Fourier series can be obtained by computation. According to the invention however the coefiicients of these developments are preferably obtained automatically by the apparatus of the invention itself. For this purpose the profile data comprise not the coefficients of the limited Fourier developments but the parametric coordinates of a certain number of points on the profile curve and the parametric values which correspond thereto. The coefficients of the Fourier developments being thus determined, the curve represented in parametric coordinates by these developments passes exactly through all of the given points but cannot coincide between such points with the theoretical curve. A better degree of coincidence is obtained of course with a larger number of data points to begin with but particularly by a proper choice of the data points along the parallel curve. The profile of a blade to be reproduced always includes portions of unequal hydrodynamic or aerodynamic importance. For example it is usually desirable that the shape in the vicinity of the leading edge conform to the pattern intended therefor more closely than is necessary at the trailing edge or at certain parts of the extrados or intrados. The number of data points being constant, it is nonetheless possible to increase the precision along particular portions of the desired profile without modifying in any way the system of parametric coordinates initially chosen. It is suificient to space the points unequally, increasing the number employed for definition of that portion of the profile which must be reproduced the most faithfully.

The apparatus of the invention comprises three principal elements:

(1) A harmonic analyzer operating by interpolation which receives as input data the numerical values of the coordinates of a number of points along the parallel curve of the desired profile and which supplies the values of the coefficients in the Fourier developments. This analyzer effects summation of products of the form:

1 E i mm) i= in ion) =1 (2) A harmonic synthesizer which receives as input data the numerical values of the Fourier coefiicients and which supplies in the form of voltages the instantaneous coordinates of the parallel curve. This synthesizer effects summation of the products:

at n-l 2a,- cos ya-I-Eb; sin jOt i= i=1 (3) Servomechanisms adjusting the machine tool to position the cutter with respect to the workpiece in accordance with these voltages.

Harmonic analyzers and synthesizers are well known in the art. See for example Harmonic Analyzer and Synthesizer by Jules Lehmann, Electronics, November 3 1949, pages 106-110 and A One-Dimensional Fourier Analog Computer by Leonid V. Azaroif, The Review of Scientific Instruments, May 1954, pages 471-477. These devices comprise essentially sinusoidal multiplying potentiometers or synchro transformers. The parameter values in the terms to be added are arranged in arithmetic progression and can be simultaneously materialized by means of shafts coupled through gearing systems having ratios arranged in arithmetic progression to a common shaft (basic shaft) whose rotation represents the variation of the parameter itself. The sum of the products can be instantaneously obtained by adding in a common resistor the output voltages of the various potentiometers or synchros. In the case of the synthesizer the parameter varies continuously from zero to 211' and the basic shaft turns continuously. In the case of the analyzer the parameter takes on a series of discrete values which are multiples of a quantity 1x and the basic shaft is successively turned to positions inclined to a zero orientation by one of these discrete values.

Sinusoidal potentiometers are well known and are available in various types. It is possible for example to use those described in US. Patent No. 2,434,057 which comprises essentially a coil mounted flat on a rectangular insulating plate and a rotating wiper whose aXis passes through the intersection of the diagonals of the plate. In synthesizers of the usual direct current and alternating current types, both coefficients and sines and cosines of the argument and of its multiples of a Fourier series are represented by current or voltage amplitudes, and the voltage representative of a term of the series is the product of the voltage representative of the coefficient and by voltage representative of the sine or cosine of the argument. In case alternating current is used, all the voltages representing the terms of the series are cophasal. The invention provides a further type of synthesizer in which the coefficients of the Fourier series are represented by amplitudes of alternating currents and the argument and its multiples are represented by the phase and the multiples thereof. Consequently the frequencies of the voltages representative of the successive terms of the series are multiples of each other. As it is convenient to give the frequency of the fundamental term the frequency of the mains, Le. 60 c./s., the cycle of this term which is the period of the series is too small for allowing the machine tool driving servomechanisms to operate. The invention provides means for deriving from the synthesizer sampling pulses having a recurrence frequency slightly differing from the series period and thus allowing the function representative of a coordinate to be stroboscopically sampled.

The invention further provides control apparatus of the type described which makes it possible for the machine tool to operate with a substantially constant output of chips selected to be suitable to the cutting tool and to the workpiece.

The invention also provides control apparatus which makes it possible for the machine to operate without vibration.

For the purpose of achieving a constant rate of cut and smooth operation the cutting speed and the rate of advance of the tool with respect to the work are controlled at each point of the motion of the cutting tool with respect to the remainder of the machine to optimum values which result in efficient operation of the machine.

According to the invention, the curve parallel to the profile is defined by points equidistant therealong, i.e. such that the curvilinear abscissa separating two successive points is constant, and the coordinates of these points are expressed as limited Fourier series as functions of a first angular parameter proportional to the curvilinear abscissa and to time. In these conditions, the movement of the cutting tool along the parallel curve would be uniform if the servomechanism operating on the traverse elements between the cutting tool and the remainder of the machine were governed by command signals in the form of voltages proportional to the coordinates thus defined. In general however for various reasons, in particular the convergence of the series, the coordinates are developed in the form of limited Fourier series of a and the first one, which varies linearly with time. The variation of the second angular parameter as a function of time is then determined by a coupling which takes into account the relation between the second parameter and the first one, which varies linearily with time. The servomechanism or other apparatus which then operates to position the cutting tool in response to voltages proportional to coordinates expressed as functions of the second angular parameter thereupon operates so that the motion of the cutting tool along the parallel curve is uniform.

It should however be noted that since in general the curvature of the profile itself is not constant, constant speed of advance of the cutting tool along its trajectory does not correspond to a constant speed of advance of the tool along the profile at the cutting point. Nonetheless the latter speed of advance does not differ greatly from the rate of advance of the cutting tool along its trajectory (e.g. of the axis of a rotating cutter) except perhaps in the vicinity of the leading and trailing edges of a blade. Since in the final analysis it is more important to obtain a constant rate of cut in the sense of a constant rate of flow of chips than to have a constant rate of advance of the tool along the profile, the outline of the raw workpiece as it comes from the foundry is so established that the thickness of material to be removed is greater in the vicinity of the leading and trailing edges of the blade than elsewhere, i.e. greater at those places where the speed of advance of the point of contact between the cutting tool and the workpiece is substantially lower than the motion of the cutting tool center along its trajectory. The invention further makes it possible to determine, on the curve parallel to the desired profile, zones within each of which the speed of motion of the cutting tool (e.g. motion of the axis of a rotating cutter with respect to the workpiece) is constant, this speed changing from zone to zone and the magnitude thereof in each zone being determined by the speed of response of the servomechanisms which position the cutting tool with respect to the workpiece.

The invention further effects determination along the parallel curve of zones within each of which the cutting speed is determined to avoid vibration of the workpiece.

The invention will now be described in detail by reference to the accompanying drawings in which:

FIGS. 1 to 3 are diagrams of right sections or profiles of diiferent types of turbine blades, showing in conjunction therewith the curves parallel to these profiles traced out by the axis of the cutting tool, indicating the geometrical significance of the angular parameter as a function of which the profile coordinates are developed.

FIGS. 4 and 5 are diagrams for comparison of a curve parallel to the theoretically desired profile section of a blade with the curve obtained by means of the apparatus of the invention, with one degree of accuracy in FIG. 4 and another degree of accuracy in FIG. 5.

FIG. 6 is a block diagram of the apparatus of the invention.

FIG. 7 is a schematic diagram of the interpolation harmonic analyzer.

FIG. 8 is a diagram of an alternate form of harmonic analyzer according to the invention.

FIG. 9 is a diagram of .a first type of harmonic synthesizer.

FIG. 10 is a diagram of an alternate form of harmonic synthesizer.

FIG. 11 is a schematic diagram of the servomechanism for positioning of the cutting tool when the curve parallel to the desired profile is defined in polar parametric coordinates.

FIG. 1&2 is a functional diagram of the harmonic synthesizer.

FIG. 13 is a diagram of a blade profile and of the curve parallel thereto useful in explaining the operation of the apparatus of the invention when the speed of advance of the cutting tool is to be constant.

FIG. 14 is a schematic diagram, partly in block form, of the apparatus of the invention as arranged for constant speed of advance of the cutting tool.

FIGS. 15 and 16 are curves representing the derivatives with respect to time of the Cartesian coordinates as a function of the angular parameter.

FIG. 17 is a diagram similar to FIG. 13 but illustrating a different distribution of the data points to be analyzed in order to reduce over certain zones of the profile to be cut the speed of response required of the servo or other machine tool driving elements.

FIG. 18 is a curve giving as a function of the angular parameter the desired cutting speed (speed of rotation of the cutter) and the cutting speed actually obtained by means of the invention.

FIG. 19 is a schematic diagram, partly in block form, of apparatus according to the invention for control of the speed of rotation of the cutting tool by means of a harmonic synthesizer.

FIG. 20 is a block diagram of a third type of harmonic synthesizer associated with sampling pulse generating means and FIG. 21 shows the sampling pulse waveform.

FIG. 22 is a diagram useful in explaining the apparatus of FIG. 23.

FIG. 23 is a block diagram useful in explaining the operation of the apparatus of the invention in a machining operation involving fractional passes or cuts and in which the harmonic analysis and synthesis of the desired parallel curve are made for successive sectors.

:FIGS. 1, 2 and 3 illustrate at reference characters 1, 3 and three light sections of blades and, at reference characters 2, 4 and 6, respectively, the curves parallel to these profiles traced out by the axis of a rotating cutter 7. This cutter may for purposes of a concrete example only be assumed to have a diameter of 5 mm. The profile 1 of FIG. 1 is a loukowski profile, that of FIG. 2 is of the type identified as No. 64-1-212 of the National Advisory Committee for Aeronautics and the profile of FIG. 3 is one comprising two circular arcs respectively of 313 and of 46 .4 cm.

One may start from the statement:

X=X Sill a In this expression X is the half distance between two tangents to the parallel curve 2, 4 or 6, these tangents being parallel to each other and containing bet-ween them the complete parallel curve as indicated in FIGS. 1-3. The angle a then represents in these figures the polar angle of a point m on the circle tangent to the two parallel tangents just mentioned, the center of this circle being at the origin of coordinates. The point m has the same abscissa as the running point M on the parallel curve. Under these conditionsdhe developments in a Fourier series limited to six terms (rt-=6) for the ordinate of these same curves are:

With respect to the curve 2 of FIG. 1 (X =39.025 mm.)

Y=-20.26 +8616 cos a 0.34 sin a +18144 cos 2a 9.13 sin 20: -12.58 cos 3a +0.90 sin 3a +2.50 cos 4a 2.69 sin 4m +3.81 cos 50: +047 sin 50: 1.24 cos 6a For the case of curve 4 of FIG. 2 (X =22.50 mm.)

Y= +5 .48 +l107-26 cos a -2.97 sin at +5.85 cos 20c 14.66 sin 20a --7.13 cos 30: +009 sin 3a +0.09 00s 40: -3.25 sin 40; +6.44 cos 5a +0.21 sin 50: -0.28 cos 60:

In the case of the curve 6 of FIG. 3, which is symmetric with respect to the axis OY of FIG. 3 (X =55 mm.)

Y=|7.02 +73.7-3 cos a +:10.15 cos 2a -8.73 cos 30c +2.53 cos 4:: +2.32 cos 502 +080 cos 6a The development of the radius vector p of the curve 2 as a function of the same parameter on is p=-+262 -4.l6 cos a --l.89 sin a 148.20 cos 2a +0.76 sin 2a --3.31 cos 3m 2.57 sin 300 -24.40 cos 4a +0.22 cos 5a -0l82 sin 5a -7.30 cos 60c More generally it is possible to write In the foregoing the coefiicients in the limited expansions have been calculated. They may however be obtained automatically by means of the apparatus of the invention from input data for the coordinates of 12 points (p=2n). The interpolation harmonic analyzer solves (for the case of the ordinate) the following equations:

Y =a +a cos a +0 cos na +171 sin (11+ +b 1 Sin (n-1)d Y =a +a cos a +a cos not;

Since as a result of the equal spacing of the parameter values ot =joz the preceding relations assume the form:

For example FIGS. 4 and 5 show in dashed lines the theoretical or desired profile 1 and the corresponding theoretical parallel curve 2, and in full lines the interpolated profiles 1 and 1" and the corresponding interpolated parallel curves 2 and 2" for the case where the parametric equations of the parallel curves are limited respectively to five and to eight terms. It will be observed that even with the parametric equations limited to five terms the coincidence between the interpolated parallel curve 2 and the theoretical curve 2 is very good and that it is nearly perfect between the interpolated curve 2" and the theoretical curve 2 when the parametric equations are limited to eight terms.

Referring to FIG. 6 the invention comprises an interpolation harmonic analyzer 81 which, from data constituted by ordinates of sample points on the curve parallel to the profile and of the parameters corresponding to these ordinates, develops the values of the coeflicients in a limited Fourier development of the ordinate, these coeflicients a to a and b,, to b,, taking for example the form of rotations of shafts 34 to 34,, and 41, to 11-1- The rotations of these shafts thus constitute input data to the harmonic synthesizer 82, which receives as supplementary input data a variable angle a in the form of a rotation of the shaft 48 representing the output of a speed reducer 47 driven by a motor 46. The synthesizer also receives as input data the quantity X again in the form of a shaft rotation. Of course, if the abscissa X of the running point M on the curve parallel to the profile, instead of being a simple sinusoidal function is like the ordinate a limited Fourier series, the terms for this series would be obtained by means of a second interpolation harmonic analyzer identical to the analyzer 81.

Output shafts 34 'to 34,, and 41, to 41,, from interpolation harmonic analyzer 81 and input shafts of the same reference numerals for harmonic synthesizer 82 are connected through clutches 90 in order to allow independent reset of said shafts.

The synthesizer 82 develops the values of X and Y as functions of oz, for example in the form of voltages. These voltages are applied respectively to abscissa and ordinate servomechanisms 87 and 88 which control screws 64 and 74 for positioning the cutting tool with respect to the workpiece in orthogonal directions of a horizontal plane.

The carriage 59 which supports the cutting tool is subjected to two orthogonal motions with respect to the workpiece 77. The first motion is derived from a motor 60 which is coupled to a speed reducer 61 and then to carriage '69 via lead screw 64. The motor 60 is energized via amplifier 62, by an error voltage which appears across resistor 63 and which is equal to the diiference between the voltage representative of X(oc) and the output voltage from potentiometer 65 whose winding is fixed to the frame of the machine tool 76 and whose slider is fixed to the carriage 69.

The second motion is applied directly to the carriage 59 which supports the cutter itself. This motion is developed by motor 70 which drives lead screw 74 through a speed reducer 71. The carriage 59 is of course coupled to the lead screw by means of a nut.

Motor 70 is energized Via amplifier 72 from an error voltage which appears across resistor 73 and which is equal to the difference between the voltage representative of Y(a) and the output voltage of potentiometer 75, whose winding is fixed to the main carriage 69 and whose slider is fixed to the secondarycarriage 59 which supports the cutting tool itself.

For simplicity, in some of the succeeding figures of the drawing, the machine tool with its servomechanisms will be represented simply by means of a rectangle 76 having four electrical terminals 109412, corresponding to those shown in FIG. 6.

The interpolation harmonic analyzer 81 is shown in FIG. 7. It includes at least p linear potentiometers numbered 17, to 17 energized from a D.C. source 8 of voltage U. On the potentiometer 17, there is set up the value of the ordinate Y, corresponding to the parameter value 0a,. Similarly potentiometer 17 is set to the value of the ordinate Y corresponding to the parameter value u The wipers 16, and 18, of potentiometer 17, are respectively connected to terminals 10, and 12, of rotary switches 9 and 11. These terminals are inclined to the rest position of the switches at an angle a, which is equal to the value of the parameter a for which the ordinate is Y,. Similarly the wipers 16,, and 18 of potentiometer 17 are respectively connected to terminals 10,, and 12,, of switches 9 and 11, and these terminals are oriented with respect to the rest position of the switches at an angle 11,, which is equal to the value of the parameter for which the ordinate is Y,,. As previously stated, the values on, to 0a,, of the parameter are equidistant, i.e. oc =joc =j21r/p and the data are the values of the ordinate Y for these linearly spaced values of the parameter (2. Consequently the terminals on switches 9 and 11 are equ-iangularly spaced. The arms 19 and 21 of switches 9 and 11 are coupled together and are connected to the voltage supply terminals of n sine and cosine multiplying potentiometers 15, to 15,,. The number n of sine potentiometers is equal to the number p of linear potentiometers. The cosine wipers 20, to 20,, of the sinusoidal potentiometers and the perpendicular sine wipers 22, to 22,, (potentiometer 15,, has no sine wiper) are mechanically coupled by a tangent screw 13 which engages with the worm wheels 14, to 14,,. The diameters of these worm wheels are such that when the wipers of potentiometer 15, rotate through an angle as the wipers of potentiometer 15 will rotate through an angle 20: and so on with the wipers of potentiometer 15,, rotating through an angle n06. Lastly the arms 19 and 21 of switches 9 and 11 are connected via a coupling 23 to the worm wheel 14, and hence rotate through the same angle as wipers 20, and 22 Shaft 13 may be turned by hand or some similar means and is designed to stop at predetermined angles 0a,, 206, noq.

When the tangent screw 13 is inclined at the angle a, to its rest position potentiometers 15, to 15,, are energized with a voltage representative of Y, and their cosine wipers supply to the terminals of load resistors 24, to 24,, voltages Y, cos a, Y, cos 211,

Y, cos not,

and their sine wipers supply to the terminals of load resistors 25, to 25,, voltages Y,sina, 171811121X].

cocoa-001i Y, sin (Ir-1),

When the tangent screw is inclined to its rest position at an angle foa the potentiometers 15, to 15,, are energized with a voltage representative of Y,- and apply to the load resistors 24, to 24,, voltages Y,- cos jot, Y,- cos 2111,

Y, cos I'ljoc,

9 and, to the load resistors 25, to 25,, voltages Y SlIl ju Y S111 2j0t Servomechanisms 26 to 26,, and 27, to 27,, have their volt-age input terminals connected respectively to the resistors 24 to 24 and 25 to 25,,

Each servomechanism includes a reference potentiometer whose winding 28 is energized from source 8 and each potentiometer also includes a wiper 29. For each servomechanism the winding 28 is mounted on a support 33 coupled to lead screw 34 This screw is driven, through a reducing mechanism 32, by a motor 30 which is energized via an amplifier 31 by means of a voltage equal to the difference between the output voltage available at wiper 20, and the output voltage of potentiometer 28.

The wiper 29 is mounted on a support 38 which is coupled to lead screw 39.- T-his screw is driven, via a speed reducer 37, by a motor 35 which is energized, via amplifier 36, by the output voltage of potentiometer 28.

A switch 40 makes it possible to apply the output voltage of potentiometer 28 (i.e. that taken at tap 29) to the terminals of resistor 24, or to the input of amplifier 36, these two positions for switch 40 being identified as 40 and 40", respectively.

With now the tangent screw 13 inclined to its rest position at an angle a and with switch 40 in position 40, the screw 34 rotates through an angle proportional to Y, cos 0: The switch 40 is then shifted to position 40" and the screw 39 turns through an angle equal to -Y, cos 0: The potentiometer 2829 is then restored to its rest position. The tangent screw 13 is then caused to shift to a position inclined to its rest position at an angle 2:1 and screw 34 rotates through an angle proportional to Y cos 20: When the switches 9 and 11 have scanned all of their terminals and 12, the shaft 34 gives an indication proportional to p EY, cos joq j=1 i.e. proportional to a, in accordance with Equation 1. In the same fashion it may be seen that the servornechanism having its input terminals connected to resistor 25 produces on its shaft 41 an indication proportional to [1,. In general terms, the servo connected to resistance 24, gives on its shaft 34, an indication pro portional to a,, and the servo connected to resistor 25, gives on its shaft 41, an indication proportional to b,.

The coefiicients a to a and b to b,, of the Fourier development of the ordinate Y limited to the order n are thus available on shafts 34 to 34, and 41, to 41,,

FIG. 8 represents a modification of the analyzer of FIG. 7. In this figure 99 99,, represent sinusoidal potentiometers of a number 2 equal to the number p of of points whose data define the curve. This number is at least equal to twice the number of terms in the Fourier development employed to represent the curve to the desired degree of accuracy. 100 100 are control rheostats which make it possible to apply to these po tentiometers supply voltages proportional to Y, Y,,. The wipers of the potentiometers are positioned by operation of a shaft 13 which drives pinions 14 14 having relative drive ratios of one, two p. 101 101 are mixing resistors, and 102 is a common adding resistor.

The voltage appearing at the terminals of the resistor 102 whose value is (i.e. which is proportional to a, when the shaft 13 occupies the position foo is applied to a servo-mechanism including amplifier 31, servomotor 30, speed reducer 32 and lead screw 34 which drives a reference potentiometer 2829. The potentiometer winding 28 is mounted on a support 33 coupled to screw 34 while the wiper 29 is stationary and is connected to resistor 102. It may be seen here that the addition of the various products which make up a given coefiicient in the Fourier series is no longer made eifeotive in a succession of operations but takes place instead in the course of a single operation, the same shaft 34 thus delivering the coefiicients a a as the shaft 13 successively takes up the discrete angular positions a, to mx Of course a second group of p sinusoidal potentiometers would on another single shaft develop the coefficients 1 to b,,

FIG. 9 illustrates diagrammatically a first synthesizer. It comprises (n+1) linear potentiometers 42 42,, having parallel dou ble windings and parallel double wipers whose wipers are coupled to shafts 34 34,, of the interpolation harmonic analyzer and an additional set of n1 linear potentiometers 43 43,, having also parallel double windings and parallel double wipers whose wipers are driven by shafts 41, 41,, All of these potentiometers are energized from the DC. source 8 whose voltage is designated U. The terminals of the two windings on the same side of the potentiometers are connected to opposite polarity terminals of source 8 so that potentials symmetrical with respect to ground will appear at the two wipers.

The output voltages from potentiometers 42 42,, energize the sinusoidal potentiometers 44 4%. Likewise the output voltages of potentiometers 43 43,, energize the sinusoidal potentiometers 45 45,,

The wipers of the sine potentiometers are driven by a motor 46 through a reducer 47, a tangent screw 48 and worm wheels 49 to 49,, having diameters such that when the wipers of sine potentiometers 44 and 45 rotate through an angle at, those of potentiometers 44, and 45, rotate through ia, the wiper of potentiometer 44,, rotating through I106.

The output voltages of the sine potentiometers which are representative of the terms a cos ice and b, sin ioc are applied to resistor 50 at the same time as the output voltage of potentiometer 44 which is representative of a Thus one obtains at the terminals of resistor 50 a voltage representative of Y(ec) A second type of synthesizer is diagrammatically illustrated in FIG. 10. It includes as set forth in the publication of Azaroff previously cited n auto transformers to 185,, which are energized by an AC. source and which in turn energize the rotors of resolvers 187 to 187 via 11 phase selector switches 186 to 186,,. Let it be supposed now that the Fourier developments are known in amplitude and phase rather than in terms of the co efiicients of the sine and cosine terms. Each autotransformer is positioned with respect to a zero position at an elongation proportional to the amplitude of the term in this series which it represents. Each rotor is positioned with respect to a zero direction at an angle equal to the phase of the term represented. The rotors are driven by a shaft rotating according to the angle oz and by gearing or coupling systems not shown such that when the first rotor turns at one speed the second turns at double speed and so on.

The first stator windings of the various resolvers are connected in parallel to the terminals of a resistor 188 at Whose terminals the voltage Y(O6) appears. Likewise the second stator windings of the resolvers are connected in parallel to the terminals of a resistor 189, at which terminals likewise there appears the voltage Y(oc) but with a quadrature phase relation with respect to the voltage available across resistor 188.

Synthesizers of the first and second types respectively illustrated in FIGS. 9 and 10 will hereinafter be diagrammatically indicated as shown in FIG. 1S2 in the form of a block including at the top thereof terminals to which are applied voltages a a a representing the cosine terms, lower terminals to which are applied the voltages b b representative of the sine terms, an input shaft rotating according to a and an output terminal on which appears the voltage Y(oc) or, more generally, the quantity represented by the Fourier series under consideration.

A third type of synthesizer 300 is illustrated in FIG. 20. This synthesizer includes a plurality of devices to generate harmonic A.C. voltages whose amplitudes are proportional to the coefficients of the Fourier terms, together with means for adding these voltages. Otherwise stated, each term of the Fourier developments which, in the synthesizer of the first and second types hitherto described was represented by a DC. or A.C. voltage of amplitude proportional both to the value of that term and to the sine or cosine of a variable angle is in the third type of synthesizer represented by an A.C. current of amplitude only proportional to the value of the coeflicient. The angular parameter a in terms of which the coordinates are developed in limited Fourier series is here represented by the phase of a reference A.C. current, i.e. the product wt of the angular frequency w and the time I.

Since the value of the angular frequency of the A.C. current which represents the fundamental in the Fourier developments is too great to permit control, by means of the voltages delivered by the synthesizer, of the servomechanisms which drive the machine tool elements for positioning the cutting tool with respect to the workpiece, a special device is provided which extends for a longer time the cycle of the voltages which represent the coordinates. If X(wt) and Y(wt) are the Cartesian coordinates of a point on the parallel curve and having a period 21r/w of the order of %0 of a second at most, the device for extending the duration of the cycle transforms these coordinates into X(r//) and YO/J) where \l/ is an angular parameter variable with time and having the desired period for example of several minutes for control of the servomechanisms.

In FIG. 20 307 designates a generator of sinusoidal si nals having for example a frequency of the order of 1000 c.p.s. The generator 307 energizes a frequency doubler 308 and a rectangular wave generator 309.

The output of the frequency doubler leads to a filter 313 which passes only the frequency 2000 cps. The output of the rectangular Wave generator 309 leads to a series of parallel connected filters 313 to 313 which pass respectively the frequencies 3000 c.p.s., 4000 c.p.s 1000n c.p.s.

Each filter is followed by two phase shifters, identified for the filter 313,, at 311 and 311 which provide quadrature outputs and which consequently produce signals representative of sine and cosine terms.

The phase shifter 311 is followed by an amplifier 31A having a gain proportional to a and the phase shifter 311 is followed by an amplifier 312 having a gain proportional to b All of the amplifier outputs are added in a load resistor 301, together With a voltage proportional to a taken from a linear potentiometer 302. Across resistor 301 there appears the voltage Y(wl).

It has been assumed that the voltages at; cos 1' wt for example, in which 1' assumes the values from zero to n, are in phase, so that the zeros of the first voltage (i=1) coincide with the zeros of the others. The parasitic phase shifts introduced by the various circuits such as the filters may falsify this assumption. To this end the phase shifters for the various harmonics of the fundamental are variable and are coupled together to produce harmonic voltages in phase with the fundamental. By means of phase comparison circuits of known type, the phase shifters 311 are coupled to the phase of the output voltage from phase shifter 311 and the phase shifters 311,-, are coupled to the phase of the output voltage of phase shifter 312 A single one of these phase comparators 306 is shown in FIG. 20, subjecting the-output voltage of phase shifter 311 to the same phase as that of the output voltage of phase shifter 311 In order to extend the cycle of the function Y from the inadequate time Z'Ir/w out to a suitable value, the said function is sampled by sampling pulses having a recurrence frequency which differs slightly from the frequency of Y and a phase rl/ which is slowly variable. The sampling pulse train is represented by expression:

sin (n+1) cot-Ill 2 This function is shown in FIG. 21. It comprises pulses 315 of high amplitude having a recurrence angular frequency of (wr//) separated by pulses of minor amplitude.

The quantity (4) is obtained 'by means of a synthesizer .303 represented by the symbol of FIG. 12 receiving at shaft 304- a slow input rotation 1p and energized with voltages representative of cos wt cos Itwt and sin wt sin not which are derived from the outputs of phase-shifters 311 to 311 and 311 to 311 of synthesizer 300.

The sampling operation which, as well known, is a multiplication of the function to be sampled by the sampling pulses, is performed in multiplying circuit 305. Multiplying circuits suitable in A.C. analog computers are Well known in the art. They are for example described at page 673 and illustrated in FIG. 19.9 of Waveforms by Chance, Hughes, McNichol, Sayre and Williams, McGraw-Hill Book Company, Inc., New York 1949.

The pulse modulated voltage issued from 305 is demodulated in a pulse demodulator which is symbolized in FIG. 20 by rectifier 314 and is used for controlling the Y co-ordinate drive of the machine.

If the parametric equations of the curve parallel to the profile are in polar rather than Cartesian coordinates (FIG. 11) the voltage representative of X sin a is developed in a simple sine potentiometer and the voltage representative of 3(a) is developed in a harmonic synthesizer. It Will hereinafter be assumed that the voltage representative of X sin a is available across resistor 93 and that the voltage representative of p(ot) is available across resistor 98. The cutting tool carriage 59 is then positioned in terms of the coordinate p just as it was in terms of the coordinate Y in the embodiment of FIG. 6, reference characteristics 50 and 70 to 75 having in FIG. 11 the same meaning as in FIG. 6.

The work piece table 92 in FIG. 11 is however mounted for rotation and is'driven by a motor 95 through a speed reducer 94. The output shaft of the reducer drives not only the table 92 but also the slider of a sine potentiometer 97 whose winding is energized with the voltage representative of p(oc). Thus it is possible to pick oif from potentiometer 97 a voltage representative of the product p sin S2 in which the S2 is the polar angle through which the table 92. isrotated. For the particularchoice made is for the perimeter a in the examples herein discussed this means:

p sin Q=X sin a If consequently the voltages ,0 sin and X 0 sin or are compared in resistor 93 one obtains an error voltage which, suitably amplified in amplifier 96 drives the servomotor 95.

A description will now be given of those features of the invention which make it possible to obtain a constant speed of advance for the cutting tool and a substantially constant fiow of chips. Referring to FIGS. 13 and 17, the profile 201 to be machined is assumed for the purposes of a concrete example to be of the type of Joukowsky. 202 is the parallel curve to be followed by the axis of the cutting tool for example a rotating miller, and 203 is the outline, in the plane of the profile 201, of the casting to be machined, reinforced in the vicinity of the leading and trailing edges of the blade for reasons already described. Let X(Ot) and Y(oc) represent generally the coordinates of a point M on the parallel curve 202 as functions of the angular parameter a and let X( and Y( 'be the coordinates of the same point as functions of an angular parameter 7 which is proportional to the curvilinear abscissa s of M measured from a point P of zero abscissa and positive ordinate, this curvilinear abscissa itself being proportional to time as already explained. 7 and s are related by in which S is the total length of the curve 202. Finally let y( be the ordinate of m as a function of the parameter X(x) and Y(0L) are as before defined by Equation 1. Similarly X( Y( and y( can be set in the form IL 21(1) =Zg r 97 +211 sin j'Y j=1 For example, there will now be given the development in series of X( and of Y( for the case where the curve 202 is harmonically analyzed for twelve points M to M which divide it into twelve equal parts:

The cutter will be uniformly translated along the parallel curve 202 if the angular parameter a is coupled to the angular parameter 7 instead of varying linearly with time and being represented by the uniform rotation of a shaft coupled to a constant speed motor. 7 itself is linear with time and is represented by the uniform rotation of a shaft, but 7 and 0: are related by the relation In FIG. 14, and 106 are two synthesizers of the type of FIG. 9 or FIG. 10. Both receive as inputs rotations from a shaft 104 whose angular position represents the parameter The synthesizer 105 receives at its input voltage terminals voltages which are representative respectively of the coefficients k k k and I l in the development of y( and its output voltage is proportional to 7( The second synthesizer 106 receives on its input voltage terminals voltages which are respectively representative of the coefiicients e e e and f f of the development of X( and its output voltage is proportional to X( v 7 The rotation on of a shaft 133 is however coupled to the rotation of shaft 104 in accordance with relation (7), in the following manner. 1'34 represents a motor and 135 a speed reducer whose output shaft 133 drives a sine potentiometer 1 36 whose two perpendicular sliders are connected to an adding network 137. This network also receives from synthesizers 105 and 106 voltages proportional to the quantities X and y( The output signal from network 137 is a voltage on conductor 138 which is a linear function of the four quantities X( 3 cos cc and sin a, and this voltage on conductor 138 goes to zero when the relation (7) is satisfied. The error voltage from network 137 is amplified in an amplifier 139 and as so amplified energizes the motor 134.

Shaft 133 provides a mechanical input to two further synthesizers .107 and 108, the first of Which receives, as electrical inputs, voltages representative respectively of c c o and of d d,, The second synthesizer 108 receives as electrical inputs voltages representative respectively of a a a and of b b These two synthesizers respectively develop output voltages representative of X(a) and of Y(OL). The output voltages from synthesizers 107 and 108 are applied respectively to subtraction networks 63 and 73 and the resulting error voltages are applied to the terminals 109 and 110 and 111 and 112 of the machine tool 76 of FIG. 6, terminals 109 and 110 being input terminals for the X drive and terminals 111 and 12 being those for the Y drive.

Correct operation of the servomechanisms for the X and Y drives of FIG. 6 comprising for the X drive amplifier 62, motor 60 and speed reducer 61 and for the Y drive amplifier 72, motor '70 and speed reducer 71 requires that the speed of variation of the corresponding Cartesian coordinate X(a) or Y( x) shall not exceed a specified value beyond which the error in machining will exceed a selected tolerance.

Speeds X( and Y( with which the coordinates X( and Y( vary are given by the following expressions which are derived respectively from relations (5) and 6) by reference to (8) X'(7)= 4.1 sin 7 +395.72 cos 7 +2.58 sin 2 +0.42 cos 2 +6.48 sin 3 --105.24 cos 3 +2.48 sin 47 2.56 cos 4 5.30 sin 5 +45.95 cos 5 1.74 sin 6 (9) Y( 149.96 sin 7 2.46 cos 7 +3566 sin 2 --18.76 cos 2 +92.49 sin 3 +4.98 cos 3 7.32 sin 4 4.04 cos 4 36.80 sin 5 +0.65 cos 5 +12.48 sin 6 The maximum permitted speed for the servos are indicated in FIGS. 15 and 16 by the horizontal lines of constant ordinate v It is seen that the speed X'( represented in absolute value by the curve 214 in FIG. 15, is everywhere below the maximum permitted value whereas the speed Y( which is represented in absolute value by the curve 205 in FIG. 16 exceeds the maximum permitted value by some 29% in the vicinity of a 7 value of 90 and by some 34% in the vicinity of a 7 value of 270. 

